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Problem 22

Use Definition 2 to find an expression for the ar…

00:47
Problem 22

Use Definition 2 to find an expression for the area under the graph of $ f $ as a limit. Do not evaluate the limit.

$ f(x) = x^2 + \sqrt{1 + 2x}, \hspace{5mm} 4 \le x \le 7 $
Problem 23

Use Definition 2 to find an expression for the area under the graph of $ f $ as a limit. Do not evaluate the limit.

$ f(x) = \sqrt{\sin x}, \hspace{5mm} 0 \le x \le \pi $
Problem 24

Determine a region whose area is equal to the given limit. Do not evaluate the limit.

$ \displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} \frac{3}{n} \sqrt{1 +\frac{3i}{n}} $
Problem 25

Determine a region whose area is equal to the given limit. Do not evaluate the limit.

$ \displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} \frac{\pi}{4n} \tan{\frac{i \pi}{4n}} $
Problem 26

(a) Use Definition 2 to find an expression for the area under the curve $ y = x^3 $ from 0 to 1 as a limit.

(b) The following formula for the sum of the cubes of the first $ n $ integers is proved in Appendix E. Use it to evaluate the limit in part (a).

$$ 1^3 + 2^3 + 3^3 + \cdots + n^3 = \biggl[ \frac{n(n + 1)}{2} \biggr]^2 $$
Amrita B.
University of California, Berkeley
Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 Problem 11 Problem 12 Problem 13 Problem 14 Problem 15 Problem 16 Problem 17 Problem 18 Problem 19 Problem 20 Problem 21 Problem 22 Problem 23 Problem 24 Problem 25 Problem 26 Problem 27 Problem 28 Problem 29 Problem 30 Problem 31 Problem 32
Problem 21

Use Definition 2 to find an expression for the area under the graph of $ f $ as a limit. Do not evaluate the limit.

$ f(x) = \dfrac{2x}{x^2 + 1}, \hspace{5mm} 1 \le x \le 3 $

Answer

$\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[\frac{2\left(1+\frac{2 i}{n}\right)}{\left(1+\frac{2 i}{n}\right)^{2}+1} \cdot \frac{2}{n}\right]$

More Answers
Frank L.
02:08
Chapter 5
Integrals
Section 1
Areas and Distances
Calculus: Early Transcendentals

Topics

Areas and Distances

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Video Transcript

Okay. The first thing we know is that Delta Axe is three minus one over ad, which is two over. And we know acts of eyes one plus items to over. And therefore, what we know is that we have the limit as UN approaches infinity starting at one like this.

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